The twentieth Fighter staff joined the eighth Air strength Command in Dec of '43, flying the P-38 in lengthy variety bomber escort position. the gang later switched over to the P-51 in July of '44. the crowd destroyed a complete of 449 enemy plane in the course of its strive against travel. Over one hundred fifty pictures, eight pages of colour, eighty pages.

The following proposition shows (under mild assumptions) that every lattice is essentially a product of irreducible lattices. Thus, the preceding example provides essentially the only way to construct reducible lattices, so most questions about lattices can be reduced to the irreducible case. Our proof relies on some results from later sections of this chapter, so it should be skipped on a first reading. 24) Proposition. Assume • G has trivial center and • Γ projects densely into the maximal compact factor of G.

In particular, if G is simple, and Z(G) = e, then G is simple as an abstract group. 6) Definition. G is semisimple if it is isogenous to a direct product of simple Lie groups. That is, G is isogenous to G1 × · · · × Gr , where each Gi is simple. 7) Remark. Semisimple groups have no nontrivial (continuous) homomorphisms to R+ (see Exercise 3), so any semisimple group is unimodular. A semisimple group, unlike a simple group, may have connected, normal subgroups (such as the simple factors Gi ). However, these normal subgroups cannot be abelian (see Exercise 1).

10(1), that if Γ \X is compact, then Q-rank(Γ ) = 0. 24 CHAPTER 2. GEOMETRIC MEANING OF R-RANK AND Q-RANK #2. 10(2), that if Γ \X is compact, then Q-rank(Γ ) = 0. Notes Helgason’s book [4] provides a thorough discussion of rank and R-rank. 10(2) was proved by B. Weiss [6]. 13 was proved by Borel and HarishChandra [1] and, independently, by Mostow and Tamagawa [5]. For nonarithmetic lattices, we will take this theorem as part of the definition of Q-rank. 16 (providing a more precise description of the geometry of the simplicial complex) was proved by Hattori [3].